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Math Help - irreducibility

  1. #1
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    irreducibility

    Find a primitive generator for \mathbb{Q}(\sqrt2,\sqrt3,\sqrt5) over \mathbb{Q}.
    I found the polynomial for the root \alpha=\sqrt2+\sqrt3<br />
+\sqrt5, which is x^8-40x^6+352x^4-960x^2+576. But I'm having trouble showing the polynomial is irreducible. Some help please.
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  2. #2
    Super Member Gamma's Avatar
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    Here is an idea

    I am not sure how to show immediately that it is irreducible; however here is another idea.

    We know that:
    [\mathbb{Q}(\sqrt2 , \sqrt3, \sqrt5):\mathbb{Q}]=[\mathbb{Q}(\sqrt2 , \sqrt3, \sqrt5):\mathbb{Q}(\sqrt2 , \sqrt3)][\mathbb{Q}(\sqrt2 , \sqrt3):\mathbb{Q}(\sqrt2)][\mathbb{Q}(\sqrt2):\mathbb{Q}]

    But it is easy to show each of the intermediary steps is a degree 2 extension with irreducible minimal polynomials x^2-5, x^2-3, x^2 - 2 all irreducible by Eisenstein with p = 5, 3, 2 respectively. So that means the total extension is:
    [\mathbb{Q}(\sqrt2 , \sqrt3, \sqrt5):\mathbb{Q}]=2*2*2=8
    Since you have found a monic polynomial of degree 8 that has this as a root it must be irreducible. Kind of roundabout, but hope it helps.
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